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📝 Collect variable terms on one side (13 MCQs)

📖 From Digital SAT Algebra • 2. Linear Equations And Inequalities • 13 questions available

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What is Collect variable terms on one side?

Definition:
Collecting variable terms on one side means using addition or subtraction to move all terms containing the variable (e.g., $x$ , $y$ , $m$ ) to a single side of the equation. This step ensures the variable appears only once, making the equation solvable.

Working:
If the equation has variables on both sides, like $5x + 3 = 2x - 1$ , subtract $2x$ from both sides to get $3x + 3 = -1$ . Now all variable terms are on the left.

Example:
Collect variable terms on one side for $8n - 5 = 3n + 10$ . Subtract $3n$ : $5n - 5 = 10$ . All $n$ -terms are now on the left. Then add 5: $5n = 15$ , divide: $n = 3$ .

Reason:
This isolation is critical because it converts a mixed equation into a simpler form where the variable can be easily solved, following the principle of maintaining balance.

Easy

Medium

Hard

📝 All Collect variable terms on one side MCQs

Q1. A student solves $7x - 3 = 5x + 9$ . Their first step is to add 3 to both sides. Is this a valid first step for isolating x? What is the most important caution they must remember after this step?

A.Yes, but they must then collect x-terms on one side, ensuring they do not lose the equality when moving the

5x

. ✅

B.No, adding 3 is wrong because it does not help collect variable terms.

C.Yes, but they must immediately divide by 7 to isolate x.

D.No, the first step must always be to subtract

5x

from both sides.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: Adding 3 is valid because it eliminates the constant term on the left. However, the core of solving is to then collect variable terms—here, subtract $5x$ from both sides to get $2x - 3 = 9$ . Many students forget to then move the $5x$ and incorrectly combine constants only.

Q2. A student writes: $4x + 5 = 2x + 1 \Rightarrow 4x + 5 - 2x = 2x + 1 - 2x \Rightarrow 2x + 5 = 1$ . Their friend says this is wrong because they should have subtracted 5 first. Which statement correctly analyzes this situation?

A.The student is correct; both orders are mathematically valid as long as variable terms are collected on one side. ✅

B.The friend is correct because you must always move constants before variables.

C.The student is wrong because they subtracted

2x

from the right side incorrectly.

D.Both are wrong because the equation has no solution.

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The student correctly applied the property of equality: subtracting $2x$ from both sides is perfectly valid. The order of operations in solving is flexible—you can move variables first or constants first, as long as you maintain balance. The friend's claim that constants must go first is a common misconception, not a mathematical rule.

Q3. A taxi company charges a flat fee of $<span class="katex-error" title="ParseError: KaTeX parse error: Can't use function '$ ' in math mode at position 5: 3.50\̲)̲ plus $" style="color:#cc0000">3.50$ plus $</span>2.25$ per mile. A competing company charges $<span class="katex-error" title="ParseError: KaTeX parse error: Can't use function '$ ' in math mode at position 5: 5.00\̲)̲ plus $" style="color:#cc0000">5.00$ plus $</span>1.75$ per mile. If you want to find the number of miles where both companies cost the same, you set $3.50 + 2.25m = 5.00 + 1.75m$ . To solve, you collect variable terms. Which of the following represents the correct equation after collecting variable terms on the left and constants on the right?

0.50m = 1.50

2.25m - 1.75m = 5.00 - 3.50

✅

4.00m = 8.50

2.25m + 3.50 = 5.00 + 1.75m

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Subtract $1.75m$ from both sides to get $3.50 + 0.50m = 5.00$ . Then subtract $3.50$ to get $0.50m = 1.50$ . This isolates the variable term. Option B is correct before simplification. Many students mistakenly add or subtract constants incorrectly; option A is the simplified result, not the collection step.

Q4. Given the equation $8 - 3t = 2t + 7$ , a student collects variable terms on the right side by adding $3t$ to both sides. Then they collect constants on the left. What is the resulting equation after these two steps?

8 = 5t + 7

8 - 3t + 3t = 2t + 7 + 3t

then

8 = 5t + 7

✅

8 - 7 = 5t

8 - 3t - 2t = 7

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Adding $3t$ to both sides cancels $-3t$ on the left, yielding $8 = 5t + 7$ . Then to isolate the constant, subtract 7 from both sides: $1 = 5t$ . Option B correctly shows the first step. Students often forget to apply the same operation to both sides or incorrectly combine like terms.

Q5. Consider the equation $5x - 8 = 2x + 10$ . Which of the following is NOT a valid first step toward solving for x?

A.Add 8 to both sides.

B.Subtract

2x

from both sides.

C.Subtract

5x

from both sides.

D.Multiply both sides by 2. ✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: Multiplying both sides by 2 is mathematically valid, but it does not help collect variable terms efficiently; it makes the equation $10x - 16 = 4x + 20$ , which is more complex. The other options directly help isolate variables or constants. This question tests whether students recognize that 'valid' in solving means helpful for isolation, not just any operation.

Q6. The equation $3(x - 2) = 2(x + 4)$ is to be solved. A student first expands to $3x - 6 = 2x + 8$ , then subtracts $2x$ from both sides to get $x - 6 = 8$ . Is this approach error-free? What is the next logical step?

A.Yes, and the next step is to add 6 to both sides to get

x = 14

. ✅

B.Yes, and the next step is to divide by

x

to get

-6 = 8/x

C.No, because subtracting

2x

from

3x

gives

x

, but they forgot to subtract

2x

from the right side.

D.No, because they should have subtracted

3x

first.

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The student correctly expanded and subtracted $2x$ from both sides. The equation $x - 6 = 8$ is correct, and adding 6 yields $x = 14$ . The common error is to forget the balance property, but here it is correctly applied. This question ensures students can verify a peer’s work and identify the proper next step.

Q7. The graph of two linear functions shows $y = 4x - 3$ and $y = x + 6$ intersecting at a point. If you set $4x - 3 = x + 6$ , after collecting variable terms on one side and constants on the other, which equation represents the horizontal distance to the intersection from the y-axis?

3x = 9

4x - x = 6 + 3

✅

x = 3

3x - 9 = 0

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: From the equation, subtract $x$ from both sides: $3x - 3 = 6$ , then add 3: $3x = 9$ . Option B is the step before simplification. The graph's intersection x-coordinate is $x=3$ . This connects algebraic collection to graphical interpretation. Distractors include simplified forms or incorrectly signed equations.

Q8. A student solves $9 - 4y = 3y + 2$ by adding $4y$ to both sides, getting $9 = 7y + 2$ . Then they subtract 2 to get $7 = 7y$ , so $y=1$ . Another student solves by subtracting $3y$ from both sides first, getting $9 - 7y = 2$ , then $7 = 7y$ . Which method is more efficient and why?

A.The second method is more efficient because it keeps the variable term positive initially. ✅

B.The first method is more efficient because it avoids negative coefficients.

C.Both are equally valid and efficient; efficiency depends on personal preference.

D.Neither is correct because the equation has no solution.

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Both methods are valid, but the first method yields $9 = 7y + 2$ then $7 = 7y$ , giving a positive coefficient. The second yields $9 - 7y = 2$ then $-7y = -7$ , requiring division by a negative. Many students prefer avoiding negatives, so method 1 is slightly more efficient for most. This encourages reflection on strategy choice.

Q9. A student claims that for the equation $2(3x - 4) = 6x - 8$ , collecting variable terms leads to $0 = 0$ . They conclude that $x$ can be any real number. Which of the following correctly evaluates this claim?

A.The claim is correct because the equation simplifies to an identity. ✅

B.The claim is incorrect because

0 = 0

means no solution.

C.The claim is incorrect because the student must have made an expansion error.

D.The claim is partially correct;

x

must be positive.

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: Expanding the left: $6x - 8 = 6x - 8$ . Subtracting $6x$ from both sides gives $-8 = -8$ , and adding 8 gives $0 = 0$ . This is an identity, true for all real numbers. This goes beyond simple collection and tests understanding of infinite solutions. Many students confuse identity with no solution.

Q10. In solving $4x + 7 = 3x - 5$ , a student writes: $4x - 3x = -5 - 7$ . Which property justifies moving $3x$ to the left and $7$ to the right as shown?

A.Subtraction property of equality (subtracting

3x

and 7 from both sides) ✅

B.Addition property of equality (adding

3x

and 7)

C.Distributive property

D.Commutative property of addition

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The student effectively subtracted $3x$ from both sides and subtracted 7 from both sides. This is the subtraction property of equality. The resulting equation $x = -12$ is correct. This question assesses whether students understand the underlying property, not just the procedure. Common misconception: thinking it's addition because of sign changes.

Q11. A rectangle's length is $3x + 2$ cm and its width is $x - 1$ cm. Its perimeter is 30 cm. The equation is $2(3x + 2) + 2(x - 1) = 30$ . After simplifying and collecting variable terms on one side, which equation correctly represents the perimeter before solving for x?

8x + 2 = 30

✅

8x = 28

6x + 4 + 2x - 2 = 30

8x - 2 = 30

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Expanding: $6x + 4 + 2x - 2 = 30$ simplifies to $8x + 2 = 30$ . Then collecting constants: $8x = 28$ . Option A is the result after collecting variable terms (all x terms are already on the left). Option B is the next step. This models a real-world geometry problem.

Q12. A student incorrectly solves $5 - 2x = 3x + 1$ by adding $2x$ to both sides to get $5 = 5x + 1$ , then subtracting 1 to get $4 = 5x$ , so $x = 0.8$ . A peer says the answer is $x = 1.2$ because they subtracted $3x$ from both sides first. Who is correct and what is the likely error of the other?

A.The first student is correct; the peer likely added instead of subtracted when moving constants. ✅

B.The peer is correct; the first student likely made an arithmetic error in division.

C.Both are correct; the equations are equivalent.

D.Neither is correct; the equation has no solution.

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: First student's steps: $5 - 2x + 2x = 3x + 1 + 2x \Rightarrow 5 = 5x + 1 \Rightarrow 4 = 5x \Rightarrow x = 0.8$ . Checking: $5 - 1.6 = 3.4$ and $3(0.8)+1 = 3.4$ , correct. The peer likely subtracted $3x$ : $5 - 5x = 1 \Rightarrow -5x = -4 \Rightarrow x = 0.8$ as well if done correctly. The peer's claim of 1.2 suggests an error in subtracting 5 or dividing. This question requires careful verification.

Q13. For the equation $\frac{2x - 1}{3} = \frac{x + 4}{2}$ , a student multiplies by 6 to get $2(2x - 1) = 3(x + 4)$ , then simplifies to $4x - 2 = 3x + 12$ . After collecting variable terms on one side, they get $x = 14$ . Which step is most critical to avoid a common sign error?

A.When subtracting

3x

from both sides, remember to subtract it from the entire left side, not just the

4x

. ✅

B.When moving the constant

-2

, change its sign to +2 correctly.

C.When multiplying by 6, ensure both numerators are multiplied correctly.

D.When expanding

3(x+4)

, remember to add 12, not multiply.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The key step is $4x - 2 - 3x = 3x + 12 - 3x \Rightarrow x - 2 = 12$ , then $x = 14$ . A common error is to only subtract $3x$ from $4x$ and forget the constant $-2$ , writing $x - 2 = 12$ is correct, but some incorrectly write $x = 12 - 2$ . This question emphasizes the importance of applying operations to entire sides.

🔗 Related Topics (MCQs)

📝 Subtraction and Addition Properties of Equality MCQs 📝 How to verify a solution to an equation MCQs 📝 Subtraction Property of Equality MCQs 📝 Addition Property of Equality MCQs 📝 Solving equations with simplification MCQs

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